Foundations of Vietoris homology theory with applications to non-compact spaces
One of the two principal objectives of the present work is to provide an underlying structure for Vietoris homology theory as it is used by topologists today. It is hoped that the existence of this structure will give formal status to the theory and enable results to be formulated and proved with a degree of precision not previously attainable. The second objective is to extend the classical theorems of Vietoris homology theory to a broader class of metric spaces by removing the restriction of compactness and replacing it by some weaker requirement. For this purpose it is necessary that the structure provided for the theory be designed in such a way that compactness is not assumed in defining the basic concepts or in establishing relationships among them. Although Vietoris' primary interest was in the mapping theorem, his methods attracted much attention. Lefschetz discussed the Vietoris theory in his 1930 Colloquium Publication [18], relating it to a homology theory of his own and to an early one of Alexandroff [l]. The Vietoris theory was soon applied to other problems by such mathematicians as Whyburn [29], [30] and Eilenberg [10]. Perhaps the most important application was that of Alexandroff who in 1932 published his celebrated paper "Dimensionstheorie. Ein Beftrag zur Geometrie der abgeschlossenen Mengen" [2]. In this paper Alexandroff constructed a homoloical theory of dimension for compact metric spaces and related the homological dimension to the covering dimension of such spaces. He was then able to characterize the covering dimension by means of a homological condition (see Chapter VI of the present work). In the process of developing the homological dimension theory, Alexandroff also developed the homology theory. His paper became and has remained to the present time the standard reference for Vietoris homology theory. Since the publication of the Alexandroff paper there have been only a few discussions of the theory itself. In his 1942 Colloquium Publication [19], Lefschetz described a general theory of algebraic topology which included Vietoris theory as well as the more recent homology theory due to Čech [8]. He was able to prove that the Vietoris homology groups are isomorphic to the corresponding Čech groups for the case of a compact metric space. Some additional discussion of the Vietoris theory occurs in Lefschetz' companion volume, Topics in Topology [20]. Slightly earlier Steenrod in constructing a homology theory for compact metric spaces [27] gave a brief comparison of his own theory to that of Vietoris. In 1950 Begle proved a Vietoris mapping theorem for topological spaces which are compact but not necessarily metrizable [3]. In doing so he used a generalized form of Vietoris cycle, first defined by Spanier [26], for which the original metric requirement was replaced by a condition described in terms of coverings of the space. In their 1952 book Foundations of Algebraic Topology [ll], Eilenberg and Steenrod gave a thorough treatment of Čech homology theory but mentioned Vietoris theory only in a note on the development of the Čech theory.
Homology theory, Algebraic topology, Metric spaces
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